SCOTTSDALE, Ariz. (AP) — A defiant Herman Cain declared Tuesday he would not drop his bid for the Republican presidential nomination in the face of allegations of inappropriate sexual behavior.

“Ain’t gonna happen,” Cain said at a news conference a day after a fourth woman accused him of unwanted sexual advances.

“We will get through this,” he added, trying to steady a campaign that has been rocked by the controversy for the past 10 days.

Cain denied anew that he had ever behaved inappropriately and said the alleged incidents “they simply didn’t happen.” He said he would be willing to take a lie detector test if he had a good reason.

Earlier in the day, Cain sought to undercut the credibility of the latest woman whose accusations are threatening his Republican presidential campaign. His chief rival, Mitt Romney, weighed in for the first time, calling the allegations “particularly disturbing.”

Cain said he called the news conference because he wanted to speak directly to the public, accusing the media of distorting his response to the allegations. He said that had never seen Sharon Bialek until she called her news conference on Monday in New York, alongside attorney Gloria Allred.

“I don’t even know who this woman is,” he said of Bialek. “I tried to remember if I recognized her and I didn’t.”

Another name confronted Cain, as well, when one of his two original accusers gave an interview to The New York Times and was identified publicly by news organizations including The Associated Press as Karen Kraushaar, now a spokeswoman in the Treasury Department’s office of inspector general for tax administration.

When asked about Kraushaar, Cain said he recalled her accusation of sexual harassment but insisted “it was found to be baseless.”

Cain contended that “the Democratic machine” was pushing the allegations but said he could not point to anyone in particular. He also suggested his accusers were lying.

Earlier, Romney, the former Massachusetts governor who has been a GOP front-runner for months, told ABC News/Yahoo! the allegations were serious “and they’re going to have to be addressed seriously.” He called the latest accusations disturbing, and Cain didn’t disagree, both in an earlier interview and at the news conference.

“He’s right. They are disturbing to me,” Cain said. “They are serious. And I have taken them seriously.”

But they’re untrue, he declared.

“I reject all of those charges,” he said, adding that “I have never acted inappropriately with anyone” and didn’t even recognize Bialek.

Cain said it was “a remote possibility” when asked if it were possible he would recall Bialek’s alleged incident in the future.

“Get all the facts in front of people, otherwise he’s going to have this continuing distraction,” Mississippi Gov. Haley Barbour, a former Republican National Committee chairman with deep ties to the GOP establishment, told MSNBC.

Though recent polling shows Cain still doing well, party operatives suggested it was only a matter of time before his political standing could suffer.

“Herman’s base is going to stick with him,” said Republican strategist Rick Tyler, Newt Gingrich’s former spokesman. “But the average Republican voter who is not as engaged as intensely in the race, is sick of this and, for Cain, the concern is they will pass on it and pass on him.”

Cain looked to keep those supporters in his corner.

“We are not going to allow Washington or politics to deny me the opportunity to represent this great nation,” he said.

“As far as these accusations causing me to back off and maybe withdraw from this presidential primary race? Ain’t gonna happen. Because I’m doing this for the American people, and the children and the grandchildren.”

Cain spoke at a hotel on the outskirts of Phoenix, where reporters crowded a small room. Outside, protesters waved signs that read: “Hey Herman. How many more women will you be calling liars?”

There were growing signs of unease in conservative circles as the Georgia businessman tried to stem the controversy in its second week.

“If there is a pattern then it’s a part of his character and then, yes, it is going to matter,” Tony Perkins, head of the conservative Family Research Center, said in an interview.

An upstart in the presidential race, Cain shot to the top of opinion polls and emerged in recent weeks as Romney’s main conservative opponent, with tea party activists and other conservatives flocking to the former pizza company executive’s tell-it-like-it-is style and outsider image.

But he’s spent the past 10 days battling accusations from women that he acted inappropriately toward them while he headed of the National Restaurant Association in the 1990s.

Cain’s campaign issued a tough statement by Tuesday about Bialek, the most recent accuser, including references to civil lawsuits in the Cook County Court system in Illinois allegedly relating to her and cited news reports of her involvement in a paternity case and a bankruptcy filing.

“In stark contrast to Mr. Cain’s four decades spent climbing the corporate ladder rising to the level of CEO at multiple successful business enterprises, Ms. Bialek has taken a far different path,” the campaign said.

It also questioned whether Bialek had a financial interest in stepping forward.

“Who is financing her legal team, have any media agreed to pay for her story, and has she been offered employment for taking these actions?”

In a round of media interviews early Wednesday, Bialek was asked repeatedly about her motives in speaking out after staying quiet for 14 years.

“I’m just doing this because it’s the right thing to do,” she said. She said she was neither paid nor offered a job to go public with her allegations. She said she waited so long to come forward because “I was embarrassed ... and I just kind of wanted it to go away.”

She said she wasn’t paying a fee to Gloria Allred, the attorney whose name has become synonymous with women’s rights issues.

At least two women who worked at the restaurant association the same time as Cain filed sexual harassment complaints with the trade group and received financial settlements.

One of them was Kraushaar.

After her name was revealed by several news sites on Tuesday, The Associated Press chose to publish her name after independently confirming that she was one of his accusers. Kraushaar and her attorney previously had attempted to keep her name out of the public discussion, but they issued an anonymous statement last week that confirmed she had complained of sexual harassment and received a financial payout from the trade group. Kraushaar later confirmed to news organizations that she had filed the complaint. And she spoke publicly on Tuesday to The New York Times, saying she had decided to speak out since her name was public.

“When you are being sexually harassed in the workplace, you are extremely vulnerable,” she said. “You do whatever you can to quickly get yourself into a job some place safe, and that is what I thought I had achieved when I left.”

Kraushaar, 55, previously worked as a news reporter, and she has held other U.S. government jobs since she left the National Restaurant Association after she settled her complaint against Cain. She also has written a children’s book with her mother-in-law, “Gas Station Charley,” about a dog. Her husband, Kevin, has worked as a lobbyist on environmental, municipal and health issues. He has donated money to both Democrats and Republicans. They live in suburban Maryland.

A third woman told The Associated Press last week that she considered filing a workplace complaint against Cain over what she deemed sexually suggestive remarks and gestures that included a private invitation to his corporate apartment. And a former pollster for the restaurant association has said he witnessed yet another episode involving a different woman.

The AP has not identified the other woman who filed a claim against Cain while working at the restaurant association. It also has not identified the third woman, who did not file a claim, because it promised confidentiality to her because she said she feared retaliation if her name became public.

Bialek came forward Monday to say that Cain, an acquaintance, groped her in car in July 1997 after they’d had dinner in Washington. Cain led the association at the time, and the unemployed Bialek was seeking job advice. She said she had been fired from her job raising money for the trade group’s education arm — told, she said, that she had not raised enough money.

Cain has been urging supporters to donate to his Iowa fund so he can meet a $999,999 goal before Wednesday’s debate in Michigan.

“The media is just itching to be able to make up more stories about how my campaign is faltering,” Cain said in an email blast to supporters.